aa r X i v : . [ m a t h . A P ] D ec UNSTABLE MANIFOLDS OF EULER EQUATIONS
ZHIWU LIN † AND CHONGCHUN ZENG ∗ Abstract.
We consider a steady state v of the Euler equation in a fixed bounded domainin R n . Suppose the linearized Euler equation has an exponential dichotomy of unstableand center-stable subspaces. By rewriting the Euler equation as an ODE on an infinitedimensional manifold of volume preserving maps in W k,q , ( k > nq ), the unstable (andstable) manifolds of v are constructed under certain spectral gap condition which is verifiedfor both 2D and 3D examples. In particular, when the unstable subspace is finite dimen-sional, this implies the nonlinear instability of v in the sense that arbitrarily small W k,q perturbations can lead to L growth of the nonlinear solutions. Introduction
We consider the incompressible Euler equation on a smooth bounded domain Ω ⊂⊂ R n , n ≥
2, under the slip (or periodic in certain directions) boundary condition(E) ( v t + ( v · ∇ ) v = −∇ p and ∇ · v = 0 x ∈ Ω v · N = 0 , x ∈ ∂ Ωwhere v = ( v , . . . , v n ) T is the velocity field and N is the unit outward normal vector of Ω.We take(1.1) W k,qEuler , { v ∈ W k,q (Ω , R n ) | ∇ · v = 0 in Ω , v · N = 0 on ∂ Ω } , q > , k > nq as the phase space. It is well known that (E) is well posed in these spaces, globally if n = 2and locally if n ≥
3. As shown below, the pressure p can be written in terms of v through aquadratic mapping.Let v be a steady solution of (E). Linearize the equation at (E) and we obtain(1.2) v t = − ( v · ∇ ) v − ( v · ∇ ) v − ∇ p , Lv where the operator L can be defined as acting only on v since the linearized pressure p can be determined by v linearly, though non-locally. To study the dynamics near v , thefirst step is to understand linear instability, that is, the spectrum of the operator L . Theproblem of linear instability of inviscid flows has a long history dated back to Rayleigh andKelvin in 19th century. But even until now, very few sufficient conditions for the existence ofunstable eigenvalues are known and most of the investigations had been restricted to shearflows and rotating flows. See [DR81],[DH66] and the references therein. Some recent resultson instability conditions can be found in [L03] [L05] for shear flows and rotating flows, andin [L04a] for general 2D flows. Besides the discrete unstable spectrum, the linearized Euleroperator may also have non-empty unstable essential spectrum due to nontrivial Lyapunovexponents of the steady flow v ([FV91] [LM91] [SL09] [LS03] [V96]). Indeed, growth of † The first author is funded in part by NSF DMS 0908175. ∗ The second author is funded in part by NSF DMS 0801319. linearized solutions can be seen in H s ( s >
1) norm near any nontrivial steady flows, due tothe stretching of the steady fluid trajectory. One also notes that the choice of the Sobolevspace (norm) actually affects the essential spectrum which corresponds to small spatial scales,but not discrete spectrum corresponding to large scales.Consider a linearly unstable steady flow v , that is, the linearized Euler operator L has anunstable discrete eigenvalue. To discuss the nonlinear instability, it is important to specifythe norms. On the one hand, certain regularity is necessary in the local well-posednessof classical solutions. On the other hand, as mentioned in the above, the choice of thenorm already affects the essential spectrum at the linear level. Moreover, even near steadystates without unstable eigenvalues, solutions are expected to grow in H s norm with s > L of nonlinear solutions is more a nonlinearreflection of the linear instability from the discrete spectrum, which also corresponds to theinstability in the large scale spatial scale (see [L04b, Section 6.2] for detailed discussions).Naturally, the ideal nonlinear instability result would be to obtain order O (1) growth in L distance (weaker energy norm) from the steady state v of solutions starting with arbitrarysmall initial perturbation from v in H s norm (stronger norm), where s > H s → L ) isnot only mathematically stronger, but also physically interesting due to the discussions inthe above.The proof of the nonlinear instability based on unstable eigenvalues is nontrivial for severalreasons. The main difficulty is that the nonlinear term v · ∇ v contains a loss of derivative.Moreover, the norm-dependent unstable essential spectrum corresponds to growth in smallspatial scales. It may interact with discrete unstable modes and then cause complications inproving nonlinear instability. In the last decade, there appeared several proofs of nonlinearinstability for Euler equations ([BGS02] [Gre00] [VF03] [FSV97] [L04b]). In [FSV97], non-linear instability in H s (cid:0) s > n (cid:1) norm was proven for n − dimensional Euler equations,under a spectral gap condition which was verified for 2D shear flows. All other papers provednonlinear instability only for the 2D case, in the more desirable H or L norms and underdifferent spectral assumptions. Particularly, in [L04b] nonlinear instability in L norm wasproved for general linearly unstable flows of 2D Euler, without additional assumption on thegrowth rate which was made in ([BGS02] [Gre00] [VF03]).When there exist a collection σ u of unstable eigenvalues of the linearized Euler operatorat the steady flow v with strictly larger real parts than the rest of the spectrum, it isvery natural from a dynamical system point of view to ask if a locally invariant unstablemanifold tangent to the eigenspace of σ u exists. The answer to this question would providea better picture of the nonlinear instability, including the dimensions and the directionsof the unstable solutions with certain minimal growth rate. More importantly, such locallyinvariant manifolds provide more precise characterization of the local dynamical pictures nearan unstable equilibria, and are also basic tools for constructing globally invariant structuressuch as heteroclinic and homoclinic orbits. These dynamical structures are important inunderstanding the turbulent fluid behaviors. The major obstacle to the construction oflocal invariant manifolds is again the loss of derivative due to the derivative nonlinearity v · ∇ v . For dissipative models such as reaction-diffusion equations ([H81]) and Navier-Stokesequations ([Yu89] [Li05]), it is rather standard to construct invariant manifolds since thedissipation terms provide strong smoothing effect to overcome the loss of derivatives in thenonlinear terms. However, for non-dissipative continuum models including Euler equations, NSTABLE MANIFOLDS 3 the linearized operators have no smoothing effect to help overcome the loss of derivativein the nonlinear terms. So it has been largely open to construct invariant manifolds forconservative continuum models such as Euler equations. In the proof of nonlinear instabilityfor 2D Euler, one takes initial data perturbed along the direction of unstable eigenfunctionsand then uses special properties of nonlinear solutions of 2D Euler to overcome the loss ofderivatives, such as the bootstrap arguments in ([BGS02] [L04b] [VF03]) and the nonlinearenergy estimates in ([Gre00]). However, these techniques can not be used for constructinginvariant manifolds, since we do not know beforehand the initial conditions for solutions onunstable or stable manifolds.In this paper, we obtain the first result on stable and unstable manifolds of Euler equationsin any dimensions. To state the precise results, we formulate the following assumptions:(A1) v ∈ W k + r,qEuler , where r ≥ ∃ λ u > λ cs >
0, and closed subspaces X u and X cs of W k,qEuler such that they satisfy L (cid:0) X u,cs ∩ domain ( L ) (cid:1) ⊂ X u,cs respectively and W k,qEuler = X u ⊕ X cs . Moreover, let L u = L | X u and L cs = L | X cs , then for some M >
0, they satisfy | e tL cs | ≤ M e λ cs t , ∀ t ≥ | e tL u | ≤ M e λ u t , ∀ t ≤ . (A3) The largest Lyapunov exponent µ (in both forward and back time) of the linearizedequations y t = Dv (cid:0) x ( t ) (cid:1) y y t = − (cid:16) Dv (cid:0) x ( t ) (cid:1)(cid:17) ∗ y, along any integral curve x ( t ) of x t = v ( x ), satisfies λ u − λ cs > K µ where K is a constant depending only on r and k .Now we give our main theorem Theorem 1.1.
Under above assumptions (A1)-(A3), there exists a unique C r − , local un-stable manifold W u of v in W k,qEuler which satisfies (1) It is tangent to X u at v . (2) It can be written as the graph of a C r − , mapping from a neighborhood of v in X u to X cs . (3) It is locally invariant under the flow of the Euler equation (E), i.e. solutions startingon W u can only leave W u through its boundary. (4) Solutions starting on W u converges to v at the rate e λt as t → −∞ for any λ <λ u − K µ .The same results hold for local stable manifold of v as the Euler equation (E) is time-reversible. Remark.
In Sections 3 and 4, the assumptions (A2)-(A3) are verified for linearly unstable D shear flows and rotating flows, as well as D shear flows. For these flows, µ = 0 andthus (A3) is automatically satisfied if an unstable eigenvalue exists. Remark.
Suppose v ∈ W k + r,q ∩ W k + r,q and the invariant decompositions W k i ,q i = X iu ⊕ X ics along with the same exponents λ u,cs satisfy (A2) – (A3) for i = 1 , . One may constructthe local unstable manifolds W ui ⊂ W k i ,q i , i = 1 , from Theorem 1.1. Assume X u = X u ,we claim W u = W u on an open neighborhood of v . In fact, let k = max { k , k } and LIN AND ZENG q = max { q , q } and W u be the unstable manifold of v in W k,q ⊂ W k i ,q i , i = 1 , . Clearly W u ⊂ W ui , i = 1 , , with the same tangent spaces and thus the above claim follows. When X u is finite dimensional, which in particular is always true in 2D under assumptionthe λ u > λ cs > ( k − µ as proved in Section 3, then the W k,q topology and L topology areequivalent on W u . Then an immediate consequence of the above theorem is the nonlinearinstability in L norm with initial data slightly perturbed from v in W k,q norm. Corollary 1.
Suppose (A1) –(A3) are satisfied and X u is finite dimensional, then there exists δ > such that there exist a solution v ( t ) such that | v (0) − v | L ≥ δ and | v ( t ) − v | W k,q → as t → −∞ exponentially. This is a stronger statement than the usual exponential nonlinear instability which bydefinition means that there exists δ > ǫ >
0, there exists a solution v ( t ) satisfying both | v (0) − v | < ǫ and sup ≤ t ≤ O ( − log ǫ ) | v ( t ) − v | ≥ δ . One notes thatthe perturbed solution v ( t ) is allowed to depend on ǫ and no condition is imposed on theasymptotic behavior of v ( t ) as t → −∞ . In the contrast, any solution v ( t ) in Corollary 1satisfies, in addition to the requirements in the nonlinear instability definition for all ǫ > v when t = −∞ and get out of the δ -neighborhood of v in L norm.In the previous works (see references above) on nonlinear instability of the Euler equation,growing solutions have usually been found in the most unstable direction of the linearizedequation (1.2) with roughly the maximal exponential growth rate. The above unstablemanifold theorem actually provides solutions growing in other relatively weaker unstable di-rections. Though not necessary, it is easier to see this when X u is finite dimensional. In fact,on the finite dimensional locally invariant manifold W u , the Euler equation (E) becomes asmooth ODE and v is a hyperbolic unstable node. When L | X u has eigenvalues with differentreal parts, one may split X u into strongly unstable subspace X uu and weaker unstable sub-space X wu such that X u = X uu ⊕ X wu . The standard invariant manifold theory implies thereexist the locally invariant weakly unstable manifold W wu tangent to X wu and the locallyinvariant strongly unstable fibers W suv with base point v ∈ W wu and extend in the directionof X uu . Those solutions in W wu grow in the directions of X wu at a slower exponential rate.Moreover, the Hartman-Grobman theorem implies that a H¨older homeomorphism on X u may transform the Euler equation restricted on W u into a linear ODE system.In Section 4, we construct linearly unstable 3D steady flow satisfying the assumptions inTheorem 1.1. By Corollary 1, this implies nonlinear exponential instability in L norm. Toour knowledge, this is the first proof of nonlinear instability of 3D Euler equation. We notethat the methods for proving nonlinear instability of 2D Euler cannot be applied to provenonlinear instability for 3D Euler. For example, the bootstrap arguments in ([BGS02] [L04b][VF03]) strongly use the fact that vorticity is non-streching in 2D and therefore do not workin 3D due to the vorticity stretching effect.Below, we sketch the main ideas in the proof of Theorem 1.1. The main difficulty inconstructing the unstable manifolds for the Euler equation lies in the fact that a derivativeloss occurs in the nonlinear terms while the linearized flow does not have the smoothingproperty. We will prove Theorem 1.1 mainly by considering the Euler equation (E) inthe Lagrangian coordinates. In a seminal paper [Ar66], V. Arnold pointed out that theincompressible Euler equation can be viewed as the geodesic equation on the group of volumepreserving diffeomorphisms. This point of view has been adopted and developed by several NSTABLE MANIFOLDS 5 authors in their work on the Euler equations, such as [EM70, Sh85, Br99, SZ08a, SZ08b] tomention a few.On the one hand, the main advantage of this approach is that (E) on a fixed domain asin this paper becomes a smooth infinite dimensional ODE [EM70] on the tangent bundle ofthe Lie group G of volume preserving diffeomorphisms of Ω and thus the difficulty arisingfrom the loss of regularity disappears. A side remark is that this is in contrast with theEuler equation with free boundaries [SZ08a, SZ08b] where the Riemannian curvature of theinfinite dimensional manifolds of volume preserving manifolds are unbounded operators andthe Euler equation can not be considered as infinite dimensional ODEs. As a clarification, bysaying that the Euler equation on fixed domains defines an ODE, we mean that it correspondsto a vector field which is everywhere defined and smooth on an infinite dimensional manifold.In this sense, evolutionary PDEs involving unbounded operators such as heat equation orwave equation do not define infinite dimensional ODEs.On the other hand, in the Lagrangian coordinates, the steady state v generates a specialgeodesic u ( t ) which coincides with the integral curve starting at the identity map of the rightinvariant vector field on G generated by v . By carefully using local coordinates along thisorbit generated by the group symmetry, we further transform the localized Euler equationinto a weakly nonlinear non-autonomous ODE with linear exponential dichotomy. Onenotes that the multiplication on this Lie group G – the composition between W k,q volumepreserving maps – is continuous, but not smooth, see Proposition 2.2. The assumption v ∈ W k + r,q ensures that the localization we choose possesses certain smoothness. Moreover,compared to the usual exponential dichotomy enjoyed by many ODEs and even PDEs (seefor example [CL88]), the exponential dichotomy here has the defect that the angle betweenthe associated invariant subspaces may not have a uniform positive lower bound in time dueto the possible growth of the linearized ODE flow of the vector field v . Moreover, this typeof unwanted growth in t may also appear in the norms of the nonlinearities. Assumption(A3) is used to overcome this non-uniformity in t of the dichotomy in the construction of theunstable manifolds via the method based on the Lyapunov-Perron integral equations. Herewe have to take the advantage of the fact that solutions on the unstable manifolds decay to v exponentially as t → −∞ . Since solutions on the center manifolds do not have similardecay properties, we still can not construct center manifolds of stead states. Remark.
The assumption (A2) is the standard linear exponential dichotomy for constructinginvariant manifolds. The extra gap assumption (A3) might be technical, but appears rathernatural in our approach. In the proof, we change back and forth between Lagrangian andEulerian coordinates, and each transformation induces a factor e µ t in the estimates. Theextra gap λ u − λ cs > K µ guarantees that after all these transformations, the exponentialdichotomy of unstable and central-stable parts still persists. The method introduced in this paper provides a general approach to construct unstablemanifolds for many other continuum models in fluid and plasmas. In these models, the lossof derivative is also due to nonlinear terms from the material derivative. By working onLagrangian coordinates, we can again overcome such a loss of derivative and the existence ofunstable manifolds is conceivable with sufficient spectral gap. We are using this approach toconstruct unstable manifolds for density-dependent Euler equations and the Vlasov-Poissonsystem for collisionless plasmas.
LIN AND ZENG Proof of Theorem 1.1
Lagrangian coordinates is a standard tool in studying the Euler equation. In Subsection2.1 and 2.2, we will present the manifold structure of the set G of the Lagrangian maps andthe ODE nature of (E) on T G . These general results have actually been proved even forEuler equations defined on Riemannian manifolds in [EM70] in a rather geometric language.However, we need to establish a more concrete framework along with more detailed estimatesto be used in the construction of local invariant manifolds in Subsections 2.3 – 2.6, which isdone in a more directly equation based manner in Subsections 2.1 and 2.2. In Subsection 2.3,we rewrite in the Lagrangian coordinates the localized Euler equation in a neighborhood ofthe solution curve generated by v and in Subsection 2.4, the linear exponential dichotomyis given. The unstable integral manifold corresponding the linear exponential dichotomyis constructed in Subsection 2.5 and then finally the unstable manifold in the Euleriancoordinates is obtained in Subsection 2.6.Throughout this section, we will use K > r and k and C > n, r, k, q, v . Both K and C may change from line to line. We willuse D or ∇ to denote the differentiation with respect to physical variables in Ω and D theFr´echet differentiations in function spaces.2.1. Lagrangian coordinates and the Lie group of volume preserving maps.
Let u ( t, · ) : Ω → Ω be the Lagrangian coordinate map defined by(2.1) u (0 , y ) = y and u t ( t, y ) = v ( t, u ( t, y )) . In particular, let u ( t, y ) be the Lagrangian map of the steady vector field v ( x ). Throughoutthe paper, we fix a constant µ > µ ≥ λ u − λ cs > K µ where µ is the Lyapunov exponent of v . From the definition of the Lyapunov exponent,we have(2.3) | u ( t, · ) | C l + | (cid:0) u ( t, · ) (cid:1) − | C l ≤ Ce lµ | t | , t ∈ R , ≤ l ≤ k + r for some C > t . This possible exponential growth of the norm of u makes the problem much more subtle than the unusual constructions of the local invariantmanifolds in differential equations.Since the flow is incompressible and k > nq , we have for any t ∈ R ,(2.4) u ( t, · ) ∈ G , { φ ∈ W k,q (Ω , R n ) | φ is a diffeomorphism, det( Dφ ) ≡ , φ ( ∂ Ω) = ∂ Ω } . Clearly the composition makes G a group. We will show that G is an infinite dimensionalsubmanifold of W k,q (Ω , R n ). This will be our configuration space when the Euler equationis written in the Lagrangian coordinates. We will work with local coordinates on G .Formally, the tangent space of G is given by(2.5) T id G = W k,qEuler , T φ G = { w | w ◦ φ − ∈ W k,qEuler } ∀ φ ∈ G , where W k,qEuler defined in (1.1) is the phase space of the velocity fields of (E). From the Hodgedecomposition, a complementary space of W k,qEuler in W k,q (Ω , R n ) is given by(2.6) ( W k,qEuler ) ⊥ = {∇ h | h ∈ W k +1 ,q (Ω , R ) } . NSTABLE MANIFOLDS 7
Here the orthogonality is in the sense that Z Ω w · ∇ hdx = 0 , ∀ w ∈ W k,qEuler , h ∈ W k +1 ,q (Ω , R ) . It is clear that both W k,qEuler and ( W k,qEuler ) ⊥ are closed subspaces of W k,q (Ω , R n ) and W k,q (Ω , R n ) = W k,qEuler ⊕ ( W k,qEuler ) ⊥ . In fact, given any X ∈ W k,q (Ω , R n ), let(2.7) w = X − ∇ h where h is the solution of(2.8) ∆ h = ∇ · X in Ω ∇ N h = X · N on ∂ Ω , then obviously X = w + ∇ h with w ∈ W k,qEuler and this verifies the direct sum. Locallynear the identity map id , we will write G as the graph of a smooth mapping from W k,qEuler to( W k,qEuler ) ⊥ and thus G is rigorously a smooth manifold. Let B δ ( · ) denote the ball of radius δ centered at 0 in the corresponding Banach space. Proposition 2.1.
There exists δ > and a smooth mapping Ψ : B δ ( W k,qEuler ) → W k,q (Ω , R n ) ,such that Ψ(0) = id , D Ψ(0) = I , Ψ( w ) − w ∈ ( W k,qEuler ) ⊥ for any w ∈ B δ ( W k,qEuler ) , and (cid:16) id + B δ (cid:0) W k,q (Ω , R n ) (cid:1)(cid:17) ∩ G ⊂ { Ψ( w ) | w ∈ B δ ( W k,qEuler ) } ⊂ (cid:16) id + B δ (cid:0) W k,q (Ω , R n ) (cid:1)(cid:17) ∩ G . Proof.
Since ∂ Ω is a smooth compact hypersurface in R n , the distance function to ∂ Ω issmooth in a neighborhood of ∂ Ω. Let d : R n → R be a smooth function with compactsupport such that it coincides with this distance function in a neighborhood of ∂ Ω. Considerthe mapping G : W k,q (Ω , R n ) → Y , { ( f, g ) ∈ W k − ,q (Ω , R ) × W k − q ,q ( ∂ Ω , R ) | Z ∂ Ω g dS = 0 } defined as G ( φ ) = (cid:0) det( Dφ ) , ( d ◦ φ (cid:1) | ∂ Ω − | ∂ Ω | Z ∂ Ω d ◦ φ dS )where | ∂ Ω | denotes the area of ∂ Ω. Obviously, G is a smooth mapping and G | G ≡ (1 , φ ∈ U and G ( φ ) = (1 ,
0) where U is a neighborhood of the identity map id in W k,q (Ω , R n ). Then φ is a diffeomorphism from Ω to its image and | φ (Ω) | = | Ω | and d ◦ φ = a , | ∂ Ω | Z ∂ Ω d ◦ φ dS imply that d ◦ φ ≡ a = 0. Otherwise a = 0 would imply that φ (Ω) either strictly covers Ω oris strictly contained in Ω, either of which contradicts with the first identity above. Therefore U ∩ G = U ∩ G − { (1 , } . It is easy to compute D G ( id ) X = ( ∇ · X, X · N − | ∂ Ω | Z ∂ Ω X · N dS ) ∈ Y, ∀ X ∈ W k,q (Ω , R n ) . LIN AND ZENG
From the standard theory of elliptic problems with Neumann boundary conditions, DG ( id ) ∇ h = (∆ h, ∇ N h − | ∂ Ω | Z Ω ∆ h dx )is an isomorphism from ( W k,qEuler ) ⊥ to Y . Therefore the proposition follows from the ImplicitFunction Theorem. (cid:3) Near any φ ∈ G , the local coordinate map Ψ composed with the right translation, Ψ( · ) ◦ φ ,gives a smooth local coordinate map near φ . Therefore G is a Banach submanifold with themodel space W k,qEuler . It is well-known [EM70] that G is not such a standard Lie group as oneneeds to be careful with the smoothness of the left translations. Let C ( φ , φ ) = φ ◦ φ andit is straightforward to verify Proposition 2.2.
C ∈ C l (cid:16)(cid:0) G ∩ W k + l,q (Ω , R n ) (cid:1) × G , G (cid:17) and, for any φ ∈ G , the righttranslation C ( · , φ ) ∈ C ∞ ( G ) . Euler equation as an ODE on T G . It is well-known that the pressure p can berepresented in terms of the velocity field v . In fact, by taking the inner product of v t and N on ∂ Ω and the divergence of the v t and using ∇ · v = 0 in Ω and N · V = 0 on ∂ Ω, we obtain(2.9) ( − ∆ p = Σ ni,j =1 ∂ i ( v j ∂ j v i ) = Σ ni,j =1 ∂ i v j ∂ j v i = tr( Dv ) in Ω ∇ N p = − N · ( v · ∇ ) v = −∇ v ( v · N ) + ∇ v N · v = v · Π( v ) on ∂ Ωwhere the symmetric operator Π ∈ C ∞ (cid:0) Ω , L ( T ∂ Ω) (cid:1) is the second fundamental form of ∂ Ωdefines as Π( x )( τ ) = ∇ τ ⊤ N with τ ⊤ ∈ T x ∂ Ω being the tangential component of τ .Based on the form of the pressure, we define the symmetric bounded bilinear mapping B : ( W k,q (Ω , R n )) → W k,q (Ω , R n ) as B ( X , X ) = ∇ γ where ( − ∆ γ = tr( DX DX ) = Σ ni,j =1 ∂ i X j ∂ j X i in Ω ∇ N γ = X · Π( X ) on ∂ Ω . The boundedness of B is clear from the standard elliptic theory. Note that here we do notassume ∇ · X , = 0 in Ω or N · X , on ∂ Ω in the definition of B . According to the Hodgedecomposition given through (2.8) and a similar calculation as in (2.9), it holds that(2.10) DX ( X ) + B ( X , X ) = ( X · ∇ ) X + B ( X , X ) ∈ W k,qEuler , ∀ X , X ∈ W k,qEuler . As a side remark, it indicates that, when embedded in L (Ω , R n ), B is the second fundamentalform of G at id which can be rigorously verified through a standard procedure.Define the mapping P : G × (cid:0) W k,q (Ω , R n ) (cid:1) → W k,q (Ω , R n ) as(2.11) P ( φ, X , X ) = B ( X ◦ φ − , X ◦ φ − ) ◦ φ. We also define the projection Q : G × W k,q (Ω , R n ) → T G as(2.12) Q ( φ, X ) = ( X ◦ φ − − ∇ h ) ◦ φ ∈ T φ G where ∇ h for X ◦ φ − is defined in (2.7). Obviously, P is symmetrically bilinear in X and X . As in (2.10), it holds(2.13) (cid:16)(cid:0) D ( X ◦ φ − ) (cid:1) ◦ φ (cid:17) ( X ) + P ( φ, X , X ) ∈ T φ G = { w : Ω → R n | w ◦ φ − ∈ W k,qEuler } . NSTABLE MANIFOLDS 9
In fact, (2.10) also leads to that, when embedded in L , P ( φ, · , · ) is the second fundamentalform of G at φ . Euler equation (E) and (2.9) imply that the Euler equation (E) takes theform in the Lagrangian coordinates(2.14) u tt + P ( u, u t , u t ) = 0 , u ( t ) ∈ G . Moreover, for any φ ∈ G , we have(2.15) P ( φ ◦ φ , X ◦ φ , X ◦ φ ) = P ( φ, X , X ) ◦ φ , i.e. P is invariant under the right translation. Therefore,(2.16) u ( t ) ◦ φ is also a solution for any solution u ( t ) of (2.14) . The following proposition states that P is a smooth mapping and thus has no regularity losswhich is far from trivial even though B is a bounded bilinear operator. To see this, one notethat even the dependence of the term X ◦ φ ∈ W k,q on φ ∈ W k,q is not smooth unless X belongs to a space of better regularity. The proof of the proposition is essentially a carefulanalysis of the commutator between the operations of B and the composition by φ ∈ G . Proposition 2.3. P : G × (cid:0) W k,q (Ω , R n ) (cid:1) → W k,q (Ω , R n ) and Q : G → L (cid:0) W k,q (Ω , R n ) (cid:1) are C ∞ and, for any m > , there exist C , K > depending only on m and k such that | ( D φ ) m P ( φ, X , X ) | W k,q ≤ C | Dφ | KW k − ,q | X | W k,q | X | W k,q (2.17) | ( D φ ) m Q ( φ, X ) | W k,q ≤ C | Dφ | KW k − ,q | X | W k,q . (2.18)Here ( D φ ) m P should be considered as a multilinear operator from T φ G to W k,q (Ω). Since P is bilinear in X , , the bounds on the derivatives of P with respect to X , follow from(2.17) for m = 0. Also, we note that | Dφ | W k − ,q ≥ k − > nq and detDφ ≡ Proof.
We will present only the proof for P as the one for Q follows through almost exactlythe same (or even slightly simpler) procedure. Due to the invariance (2.15) of P underthe right translation, we only need to show its smoothness near φ = id . As φ belongs tothe manifold G , the smoothness of P is equivalent to the smoothness of P (cid:0) Φ( w ) , X , X (cid:1) with respect to w ∈ W k,qEuler and X , ∈ W k,q (Ω , R n ) for any smooth local coordinate mapΦ : B δ ( W k,qEuler ) → G . Moreover, to prove the Fr´echet smoothness of P , it suffices to showthe Gˆateaux differentiability of P up to the m -th order for any m >
0, which would implythat Gˆateaux derivative D m − P is continuous and thus it is also the ( m − m -th order, it suffices to provethe smoothness of P (cid:0) φ ( s , . . . , s m ) , X ( s , . . . , s m ) , X ( s , . . . , s m ) (cid:1) ∈ W k,q (Ω , R n )for any φ ( s , . . . , s m ) ∈ G ⊂ W k,q (Ω , R n ) and X , ( s , . . . , s m ) ∈ W k,q (Ω , R n ) with smoothparameters ( s , . . . , s m ) ∈ U ⊂ R m . We will show by induction(ML) Z , ∂ s . . . ∂ s m P ( φ, X , X ) | s = ... = s m =0 ∈ W k,q (Ω , R n )and obtain its bound in the form of (2.17).The boundedness of the bilinear transformation B implies (2.17) and (ML) for m = 0.Assume (ML) and (2.17) hold for 0 ≤ m < m and we will prove it for m = m . Let P ( φ, X , X ) = ( ∇ γ ) ◦ φ , where γ , defined as in the definition of B , depends on s , . . . , s m through φ , X , and X . In the following, since it will be much easier to carry out somecalculations in the Eulerian coordinates, let˜ X j = X j ◦ φ − , j = 1 , τ i = ( ∂ s i φ ) ◦ φ − , D s i = ∂ s i + ∇ τ i , i = 1 , . . . , m and ˜ Z = D s . . . D s m ∇ γ = Z ◦ φ − . Clearly, it is sufficient to show ˜ Z ∈ W k,q , which will be achieved by studying its normalcomponent on ∂ Ω, divergence, and curl.To start, for any vector field W ∈ W k − m ,q (Ω , R n ), let Y ( y ) = (cid:0) Dφ ( y ) (cid:1) − W (cid:0) φ ( y ) (cid:1) , orequivalently Dφ ( Y ) = W ◦ φ , then Dφ ( ∂ s i Y ) = − Dφ s i ( Y ) + (cid:0) ( DW ) ◦ φ (cid:1) φ s i = ⇒ ∂ s i Y ∈ W k − ,q (Ω , R n ) . Differentiating the above identity one more time implies Dφ ( ∂ s j s i Y ) = − Dφ s j ( ∂ s i Y ) − Dφ s i ( ∂ s j Y ) − Dφ s j s i ( Y )+ (cid:0) ( DW ) ◦ φ (cid:1) φ s j s i + (cid:0) ( D W ) ◦ φ (cid:1) ( φ s j , φ s i )and thus the smoothness of φ and W yield ∂ s j s i Y ∈ W k − ,q (Ω , R n ) if m ≥ . Repeating this procedure we obtain ∂ s i ...s im Y ∈ W k − ,q (Ω , R n ) inductively. Changing fromthe Eulerian coordinates to the Lagrangian coordinates, we have (cid:0) ∇ W ˜ Z − D s . . . D s m ( D γ ( W )) (cid:1) ◦ φ = ∇ Y Z − ∂ s . . . ∂ s m ∇ Y (cid:0) ( ∇ γ ) ◦ φ (cid:1) . Applying the induction assumption to the last term above and using the commutator formula[ ∂ s i , ∇ Y ] = ∇ ∂ si Y to move the ∇ Y to the outside to produce ∇ Y Z , we obtain(2.19) (cid:0) ∇ W ˜ Z − D s . . . D s m ( D γ ( W )) (cid:1) ◦ φ = ∇ Y Z − ∂ s . . . ∂ s m ∇ Y (cid:0) ( ∇ γ ) ◦ φ (cid:1) ∈ W k − ,q . Taking W = e , . . . , e n of the standard basis of R n , (2.19) and the definition of P implythe curl ∇ × ˜ Z contains only the commutators terms and thus satisfies ∇ × ˜ Z ∈ W k − ,q . Similarly the divergence satisfies ∇ · ˜ Z + D s . . . D s m (tr( D ˜ X )( D ˜ X )) ∈ W k − ,q . Expanding the above last term using the product rule on D s i , it consists of terms in theform of( D s i . . . D s im ∂ l ˜ X l )( D s j . . . D s jm − m ∂ l ˜ X l ) , { i , . . . , i m , j , . . . j m − m } = { , . . . , m } . Move ∂ l and ∂ l to the outside in the same fashion as in the derivation of (2.19) (replacing W by e l , and ∇ γ by ˜ X l , , ) and using the smoothness of X , in s , . . . , s m , it is easy toobtain D s . . . D s m (tr( D ˜ X )( D ˜ X )) ∈ W k − ,q and thus ∇ · ˜ Z ∈ W k − ,q . Finally, using D s i N = Π( τ i ) , D s j D s i N = ( ∇ τ j Π)( τ i ) + Π( φ s j s i ◦ φ − ) , . . . NSTABLE MANIFOLDS 11 the induction assumption, and the assumption N · ∇ γ = 0 on ∂ Ω in the definition of P it isstraight forward to obtain˜ Z · N = N · D s . . . D s m ∇ γ ∈ W k − q ,q ( ∂ Ω , R )in the same fashion. Therefore, the standard estimates in elliptic theory implies that (ML)holds for m = m . Moreover, inequality (2.17) follows from the observation that the compo-sition by φ or φ − only produces terms like | φ | lW k,q in the estimates of the W k,q norms andthus the proof of the proposition is complete. (cid:3) Proposition 2.3 provides the key element for us to prove that (2.14) is a smooth secondorder ODE on the infinite dimensional configuration manifold G . Proposition 2.4.
For any u ∈ G and w ∈ T u G , the initial value problem of the Eu-ler equation (2.14) has a unique solution (cid:0) u ( t ) , u t ( t ) (cid:1) ∈ T G , locally in time, depending on ( t, u , w ) smoothly.Proof. Due to the right translation of (2.14) given in (2.16), we may assume u belongs to asmall neighborhood of id . From Proposition 2.1,( w, ∇ h ) → Ψ( w ) + ∇ h is a local diffeomorphism from W k,qEuler × ( W k,qEuler ) ⊥ to W k,q (Ω , R n ). Taking the Ψ( w ) compo-nent, we obtain a smooth Φ : id + B δ (cid:0) W k,q (Ω , R n ) (cid:1) → G such that(2.20) u − Φ( u ) ∈ ( W k,qEuler ) ⊥ , ∀ u ∈ id + B δ (cid:0) W k,q (Ω , R n ) (cid:1) and Φ( u ) = u, ∀ u ∈ G ∩ (cid:16) id + B δ (cid:0) W k,q (Ω , R n ) (cid:1)(cid:17) . Consider a modification of (2.14)(2.21) u tt + P (cid:0) Φ( u ) , u t , D Φ( u ) u t (cid:1) = 0 , u ∈ B δ (cid:0) W k,q (Ω , R n ) (cid:1) u t ∈ B δ (cid:0) W k,q (Ω , R n ) (cid:1) . From the smoothness of Φ and P , equation (2.21) is a smooth ODE defined on an opensubset of an infinite dimensional Banach space (cid:0) W k,q (Ω , R n ) (cid:1) and thus is locally well-posedwith smooth dependence on the initial value. Moreover, any solution u ( t ) of (2.21) satisfying u ( t ) ∈ G for all t also solves (2.14). Therefore, to complete the proof, we only need to showthat, if the initial data is given on T G , then the solution of (2.21) also stays on T G , i. e. u ( t ) ∈ G and u t ( t ) ∈ T u ( t ) G . Let v = u t ◦ (cid:0) Φ( u ) (cid:1) − . Equation (2.21) yields v t ◦ Φ( u ) + (cid:0) ( Dv ) ◦ Φ( u ) (cid:1)(cid:0) D Φ( u ) u t (cid:1) = u tt = −P (cid:0) Φ( u ) , u t , D Φ( u ) u t (cid:1) and thus (2.13) implies v t ◦ Φ( u ) ∈ T Φ( u ) G or equivalently v t ∈ T id G = W k,qEuler . Therefore v ( t ) ∈ W k,qEuler , and thus u t = v ◦ Φ( u ) ∈ T Φ( u ) G , for all t follows from the initial assumption.Consequently (cid:0) u − Φ( u ) (cid:1) t = (cid:0) I − D Φ( u ) (cid:1) u t = 0where we also used that (2.20) implies D Φ( u ) X = X for any X ∈ T Φ( u ) G . Therefore u = Φ( u ) ∈ G and the proof completes. (cid:3) According to (2.16), the second order ODE (2.14) defined on the Lie group G is invariantunder the right translation. The standard procedure of taking v = u t ◦ u − reduces it to afirst order equation on the corresponding Lie algebra W k,qEuler = T id G , which turns out to bethe usual form (E) of the Euler equation in the Eulerian coordinates. However, according to Proposition 2.2, the composition as the multiplication operation on the group G is notsmooth, this procedure induces the loss of one order of spatial derivative and make (E) intoa PDE, i. e. the right side of (E) does not define a smooth vector field on W k,qEuler .2.3. Euler equation near v as a non-autonomous ODE on T G . Even though Eulerequation is equivalent to an infinite dimensional ODE on T G , the Lagrangian frame workalso brings two complications: • G is not flat, which means that we may have to carry out the analysis in localcoordinates on G and • the steady velocity field v ( x ) of the Euler equation (E) corresponds to a dynamicsolution (cid:16) u ( t, y ) , u t ( t, y ) = v (cid:0) u ( t, y ) (cid:1)(cid:17) of (2.14).It is natural to look for ways to take the advantage of the group structure of G to reduce(2.14), localized near u ( t ), to a non-autonomous second order ODE defined in local coor-dinate neighborhood of id . Based on the comments at the end of Subsection 2.2, taking w = u ◦ u − for u ( t ) near u ( t ) would result in loss of regularity, which can also be seen ex-plicitly through the simple calculation u t = w t ◦ u + ( Dw ◦ u )( v ◦ u ) leading to w t / ∈ W k,q due to the presence of Dw . Instead, for any solution (cid:0) u ( t ) , u t ( t ) = v ( t ) ◦ u ( t ) (cid:1) ∈ T G of theEuler equation (2.14) with u ( t ) close to u ( t ), let(2.22) u = Φ( t, w ) , u ( t ) ◦ Ψ (cid:0) w ( t ) (cid:1) , w ( t ) ∈ B δ ( W k,qEuler ) , where Ψ is given in Proposition 2.1 which also implies Φ( t, · ), for any t ∈ R , is a localdiffeomorphism from W k,qEuler to G . Therefore, the linearizations˜ X = D Φ( t, w ) X = (cid:0) Du ◦ Ψ( w ) (cid:1) D Ψ( w ) X, X ∈ W k,qEuler X = (cid:0) D Φ( t, w ) (cid:1) − ˜ X = (cid:0) D Ψ( w ) (cid:1) − (cid:0) ( Du ) − ◦ Ψ( w ) (cid:1) ˜ X, ˜ X ∈ T u G are isomorphisms from between W k,qEuler and T u G . Substitute (2.22) in to (2.14), one maycompute u t = Φ t ( t, w ) + D Φ( t, w ) w t = u t ◦ Ψ( w ) + ( Du ◦ Ψ( w )) D Ψ( w ) w t (2.23) v = u t ◦ u − = v + (cid:0) ( Du ) ◦ u − (cid:1) (cid:0) ( D Ψ( w ) w t ) ◦ Ψ( w ) − ◦ u − (cid:1) (2.24)and u tt =Φ tt ( t, w ) + 2 D Φ t ( t, w ) w t + D Φ( t, w ) w tt + D Φ( t, w )( w t , w t )= u tt ◦ Ψ( w ) + 2 ( Du t ◦ Ψ( w )) D Ψ( w ) w t + (cid:0) D u ◦ Ψ( w ) (cid:1) (cid:0) D Ψ( w ) w t , D Ψ( w ) w t (cid:1) + ( Du ◦ Ψ( w )) (cid:0) D Ψ( w )( w t , w t ) + D Ψ( w ) w tt (cid:1) . Therefore, for u ( t ) close to u ( t ), the Euler equation (2.14) is rewritten as(2.25) w tt + F ( t, w, w t ) = 0 , w ∈ B δ ( W k,qEuler ) , w t ∈ W k,qEuler , where, for w ∈ B δ ( W k,qEuler ) and X ∈ W k,qEuler , the term F ( t, w, X ) is arranged into the linearand quadratic parts in X (2.26) F ( t, w, X ) = A ( t, w ) X + B ( t, w )( X, X ) ∈ W k,qEuler NSTABLE MANIFOLDS 13 with the linear and bilinear (in X ) operators A ( t, w ) and B ( t, w ) are given by A ( t, w ) X =2 (cid:0) D Φ( t, w ) (cid:1) − (cid:0) D Φ t ( t, w ) X + P ( u, v ◦ u, ˜ X ) (cid:1) =2 (cid:0) D Ψ( w ) (cid:1) − (cid:0) ( Du ) − ◦ Ψ( w ) (cid:1)(cid:0) ( Dv ◦ u ) ˜ X + P ( u, v ◦ u, ˜ X ) (cid:1) (2.27) B ( t, w )( X, X ) = (cid:0) D Φ( t, w ) (cid:1) − (cid:0) D Φ( t, w )( X, X ) + P ( u, ˜ X, ˜ X ) (cid:1) = (cid:0) D Ψ( w ) (cid:1) − (cid:0) ( Du ) − ◦ Ψ( w ) (cid:1)(cid:16)(cid:0) Du ◦ Ψ( w ) (cid:1) D Ψ( w )( X, X )+ (cid:0) D u ◦ Ψ( w ) (cid:1)(cid:0) D Ψ( w ) X, D Ψ( w ) X (cid:1) + P ( u, ˜ X, ˜ X ) (cid:17) , (2.28)where u = Φ( t, w ) = u ◦ Ψ( w ) ˜ X = D Φ( t, w ) X = (cid:0) Du ◦ Ψ( w ) (cid:1) D Ψ( w ) X. In the above calculation, the invariance of P under the right translation (2.15) and equa-tion (2.14) were used in handling both u tt and u tt . Here even though the linear operator (cid:0) D Φ( t, w ) (cid:1) − acts only on the subspace T u G , the terms A ( t, w ) X and B ( t, w )( X, X ) arewell-defined and thus F ( t, w, X ) ∈ W k,qEuler . In fact, (2.13) implies D Φ t ( t, w ) X + P ( u, v ◦ u, ˜ X ) = ( Dv ◦ u ) ˜ X + P ( u, v ◦ u, ˜ X ) ∈ T u G and thus A ( t, w ) is well-defined. To see that B ( t, w ) is well-defined, we note the secondlinearization of Φ along a line w + sX , s ∈ R , at s = 0, is given by u ss = D Φ( t, w )( X, X ) = (cid:0) Du ◦ Ψ( w ) (cid:1) D Ψ( w )( X, X )+ (cid:0) D u ◦ Ψ( w ) (cid:1)(cid:0) D Ψ( w ) X, D Ψ( w ) X (cid:1) . Let Y = u s ◦ u − = ˜ X ◦ u − ∈ W k,qEuler , then u ss = Y s ◦ u + DY ( ˜ X ) . Since Y s ∈ W k,qEuler and (2.13) implies DY ( ˜ X ) + P ( u, ˜ X, ˜ X ) ∈ T u G , we obtain D Φ( t, w )( X, X ) + + P ( u, ˜ X, ˜ X ) = u ss + P ( u, ˜ u s , ˜ u s ) ∈ T u G and thus B ( t, w )( X, X ) = (cid:0) D Φ( t, w ) (cid:1) − (cid:0) D Φ( t, w )( X, X ) + P ( u, ˜ X, ˜ X ) (cid:1) ∈ W k,qEuler is well-defined. Remark.
An alternative way to rewrite the Euler’s equation to derive the above form is tofollow the Lagrangian variational principle. One may first express the action
R R Ω | u t | dydt ,defined on T G , using (2.22) and (2.23) . The Euler’s equation in terms of w follows from thethe variation of the action. Recall we assumed in (A1) that v ∈ W k + r,q with r ≥ µ > µ ≥ Lemma 2.5.
The nonlinear mapping F satisfies F ∈ C r − (cid:0) R × B δ ( W k,qEuler ) × W k,qEuler , W k,qEuler (cid:1) , F ( t, w, ≡ . Moreover, there exist
K > depending only on r and k and C > depending only on r, n, k, q, v , such that, for t ∈ R , w, X ∈ W k,qEuler and | w | W k,q < δ , we have | A ( t, · ) | C r − (cid:0) B δ ( W k,qEuler ) ,L ( W k,qEuler ) (cid:1) ≤ Ce Kµ | t | , | B ( t, · ) | C r − (cid:0) B δ ( W k,qEuler ) ,L ( W k,qEuler ⊗ W k,qEuler ,W k,qEuler ) (cid:1) ≤ Ce Kµ | t | . Here the highest order derivative is in the Gˆateaux sense which is sufficient to yield the C r − , bounds.Proof. The smoothness of F follows directly from its expression (2.26) – (2.28) and Propo-sitions 2.1 – 2.4. The property F ( t, w, ≡ u ◦ φ, u t ◦ φ ) is a solution of (2.14) for any φ ∈ G . Then (2.22)implies that, for any w ∈ B δ ( W k,qEuler ), ( w,
0) is a time independent solution of (2.25) andthus F ( t, w,
0) = 0. The derivation of the estimates is tedious, but straightforwardly fromProposition 2.1 – 2.3, (2.17), and (2.26) – (2.28). (cid:3)
Remark.
As proved in Proposition 2.3, (2.14) is a smooth infinite dimensional ODE. How-ever, the local coordinate systems based on the composition would always cause loss of deriva-tives due to Proposition 2.2. Here our local coordinate mapping Φ( t, · ) allows us to obtainsome limited smoothness due to assumption (A1) of the extra regularity of v . Linear exponential dichotomy in Lagrangian coordinates.
Since F ( t, w,
0) = 0for any small w ∈ T id G , we have D w F ( t, w,
0) = 0. We can rewrite (2.25) as(2.29) z t = A ( t ) z + F ( t, z ) , z = ( z , z ) T ∈ B δ ( W k,qEuler ) × W k,qEuler where A ( t ) = (cid:18) I − A ( t, (cid:19) , F ( t, z ) = (cid:18) A ( t, z − F ( t, z , z ) (cid:19) . From (2.27), the explicit form of A ( t,
0) is given by A ( t, X = 2 (cid:0) Du ( t ) (cid:1) − (cid:16)(cid:0) Dv ◦ u ( t ) (cid:1) Du ( t ) X + P (cid:0) u , v ◦ u ( t ) , Du ( t ) X (cid:1)(cid:17) and Lemma 2.5 implies that there exists C > δ such that(2.30) F ( t,
0) = 0 = D z F ( t, , |D z F ( t, · ) | C r − (cid:0) B δ ( W k,qEuler ) ,W k,qEuler (cid:1) ) ≤ Ce Kµ | t | , where again the highest order derivative is in the Gˆateaux sense which is sufficient to yieldthe C r − , bounds. The linearization of (2.25) takes the form of(2.31) w tt + D X F ( t, , w t = 0 ⇔ z t = A ( t ) z whose well-posedness is guaranteed by Lemma 2.5. Let T ( t, t ) be the solution operator of(2.31) with initial time t and terminal time t . NSTABLE MANIFOLDS 15
On the one hand, for w ∈ W k,qEuler , z = ( w, T is a solution of (2.31). On the other hand,linearizing (2.24) at the steady solution z = ( w , T one may compute z = ( w ( t ) , w t ( t )) T isa solution of the linearization of (2.29) at z = ( w , T , where w t ( t ) = (cid:0) D Ψ( w ) (cid:1) − (cid:0) ( Du ( t )) − ◦ Ψ( w ) (cid:1)(cid:0) v ( t ) ◦ u ( t ) ◦ Ψ( w ) (cid:1) and v = v ( t ) is a solution of (1.2). In particular, taking w = 0, we have that z =( w ( t ) , w t ( t )) T is a solution solution of (2.31) where(2.32) w t ( t ) = (cid:0) Du ( t ) (cid:1) − (cid:0) v ( t ) ◦ u ( t ) (cid:1) . This correspondence between the linearized solutions of (1.2) and (2.31) and assumption(A2) would yield the ( t -dependent) exponential dichotomy ( W k,qEuler ) = Y u ( t ) ⊕ Y cs ( t ) of (2.31)such that T ( t, t ) Y u,cs ( t ) = Y u,cs ( t ) and the exponential decay rate of T ( t, t ) as t → −∞ (or+ ∞ ) in Y u ( t ) (or Y cs ( t )) is bounded roughly by λ u (or λ cs ). To define Y u ( t ), it is natural from(2.32) that, for ( w, w t ) ∈ Y u ( t ), w t takes the form of (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v ◦ u ( t ) (cid:1) with v ∈ X u ,where L (as well as L u,cs ) is the linear operator defined in the linearized Euler equation (1.2).Also the decay of w as t → −∞ requires it take the form of w ( t ) = R t −∞ w t dt ′ . Therefore,for any t ∈ R , let Y u ( t ) = { (cid:16) Z t −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v ) ◦ u ( τ ) (cid:1) dτ, (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v ◦ u ( t ) (cid:1)(cid:17) T | v ∈ X u } . The convergence of the above infinite integral follows directly from assumptions (A2) and(A3) with K sufficient large depending only on k . Similarly, for ( w, w t ) ∈ Y cs ( t ), w t takesthe form of (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v ◦ u ( t ) (cid:1) with v ∈ X cs , and w should take the form of w + Z t (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e tL v ) ◦ u ( τ ) (cid:1) dτ. However, w ∈ W k,qEuler is arbitrary and thus we can absorb the integral term into w . Define Y cs ( t ) = { (cid:16) w, (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v ◦ u ( t ) (cid:1)(cid:17) T | w ∈ W k,qEuler , v ∈ X cs } . Lemma 2.6.
It holds ( W k,qEuler ) = Y cs ( t ) ⊕ Y u ( t ) . Moreover, let P u,cs ( t ) ∈ L (cid:0) ( W k,qEuler ) (cid:1) bethe projections associate to this decomposition and T u,cs ( t, t ) = T ( t, t ) | Y u,cs ( t ) . Then for any t, t ≤ , we have T u,cs ( t, t ) Y u,cs ( t ) = Y u,cs ( t ) and there exist constants K > depending only on k and C ≥ depending only on k, n, q, v such that if λ u > Kµ , we have | P u,cs ( t ) | L (cid:0) ( W k,qEuler ) (cid:1) ≤ C e − Kµt , ∀ t ≤ | T cs ( t, t ) | ≤ C e λ cs ( t − t ) − Kµt , ∀ ≥ t ≥ t | T u ( t, t ) | ≤ C e ( λ u − Kµ )( t − t ) − Kµt , ∀ t ≤ t ≤ . Remark.
This lemma shows that unlike the traditional exponential dichotomy, the norms ofthe projections in the invariant splitting here is not uniformly bounded in t and may approach ∞ as t → −∞ . This means that the angles between the unstable and center-stable subspaces Y u,cs ( t ) of (2.31) may not have a uniform positive lower bound as t → −∞ . Proof.
We first show the invariance of Y cs,u ( t ) under T ( t, t ). In fact, for any Y u ( t ) ∋ z = ( w, w ) T = (cid:16) Z t −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v ) ◦ u ( τ ) (cid:1) dτ, (cid:0) Du ( t ) (cid:1) − (cid:0) e t L v ◦ u ( t ) (cid:1)(cid:17) T , where v ∈ X u , let v ( t ) = e tL v ∈ X u , ˜ w = Z −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v ) ◦ u ( τ ) (cid:1) dτ. Clearly, Y u ( t ) ∋ z ( t ) = (cid:0) w ( t ) , w ( t ) (cid:1) T , (cid:16) Z t −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v ) ◦ u ( τ ) (cid:1) dτ, (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v ◦ u ( t ) (cid:1)(cid:17) T =( ˜ w, T + (cid:16) Z t (cid:0) Du ( τ ) (cid:1) − (cid:0) v ( τ ) ◦ u ( τ ) (cid:1) dτ, (cid:0) Du ( t ) (cid:1) − (cid:0) v ( t ) ◦ u ( t ) (cid:1)(cid:17) T is the solution of (2.31) with initial data ( ˜ w, v ) at t = 0. Moreover, it satisfies z ( t ) = z andthus Y u ( t ) ∋ z ( t ) = T ( t, t ) z which implies the invariance of Y u ( t ).The above arguments also leads to the decay estimate of T u ( t, t ). In fact,one may compute w ( t ) = (cid:0) Du ( t ) (cid:1) − (cid:16)(cid:0) e ( t − t ) L (( Du ( t ) w ) ◦ ( u ( t )) − ) (cid:1) ◦ u ( t ) (cid:17) . Since v ∈ X u , we obtain from (2.3) and assumptions (A2) and (A3) | w ( t ) | W k,q ≤ Ce λ u ( t − t ) − Kµt | w | W k,q , ∀ t ≤ t ≤ . Finally from w ( t ) = R t −∞ w ( τ ) dτ , we obtain the estimate for T u ( t, t ). The proof of theinvariance of Y cs ( t ) and the estimate for T cs ( t, t ) are similar.To prove the direct sum and obtain the bounds on the projection operators, let P cs,u ∈ L ( W k,qEuler ) be the projections given by the decomposition W k,qEuler = X cs ⊕ X u assumed inhypothesis (A2). Given any z = ( w, w ) T ∈ ( W k,qEuler ) and t ≤
0, let v cs,u ( t ) = P cs,u e − tL (cid:16)(cid:0) Du ( t ) w (cid:1) ◦ u ( t ) − (cid:17) ∈ X cs,u w ( t ) = w − Z t −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v u ( t )) ◦ u ( τ ) (cid:1) dτ From (2.3) and assumptions (A2) and (A3), we have | e τL v u ( t ) | W k,q ≤ Ce λ u ( τ − t ) − Kµt | w | W k,q , ∀ τ ≤ t. Let z u = (cid:16) Z t −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v u ( t )) ◦ u ( τ ) (cid:1) dτ, (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v u ( t ) ◦ u ( t ) (cid:1)(cid:17) T ∈ Y u ( t ) z cs = (cid:16) w ( t ) , (cid:0) Du ( t ) (cid:1) − (cid:0) e tL v cs ( t ) ◦ u ( t ) (cid:1)(cid:17) T ∈ Y cs ( t ) . Obvious z = z u + z cs and this splitting is unique. Therefore ( W k,qEuler ) = Y cs ( t ) ⊕ Y u ( t ) and P cs,u ( t ) z = z cs,u . It is straight forward to first obtain the estimates on P u ( t ) based on theabove inequalities and the bound on P cs ( t ) = I − P u ( t ) also follows. (cid:3) NSTABLE MANIFOLDS 17
Let F u,cs ( t, z ) = P u,cs ( t ) F ( t, z ) . Then Lemma 2.6 and (2.30) imply that there exist
K > C > t ≤
0, it holds(2.33) F cs,u ( t,
0) = 0 = D z F cs,u ( t, , |D z F u,cs ( t, z ) | C r − ( B δ ( W k,qEuler ) ,W k,qEuler ) ≤ C e − Kµt , where again the highest order derivative is in the Gˆateaux sense which is sufficient to yieldthe C r − , bounds.2.5. Integral unstable manifolds of (2.25) . We will follow the Lyapunov-Perron integralequation method to construct the integral unstable manifolds. In the standard constructionof invariant manifolds, where the bounds on the invariant splitting and the nonlinear termsare uniform in t , small Lipschitz constant of F cs,u is sufficient in the construction of localinvariant manifolds. As we do not have t -uniform estimates here, we repeatedly used theproperty that F ( t, z ) = O ( | z | ) to yield an extra | z ( t ) | = O ( e λt ) with λt <
0. This quadraticnature of F combined with the exponential gap condition (2.36) allows us to complete theproof of Proposition 2.7 in the below. However, similar construction would not work for theconstruction of the center-stable manifold since solutions z ( t ) on the center-stable manifolddo not satisfy z ( t ) → t → ±∞ .For λ ∈ ( λ cs , λ u ) and δ ≤ δ to be determined, let Γ = { z = ( z u , z cs ) ∈ C (cid:0) ( −∞ , , B δ ( W k,qEuler ) (cid:1) | | z | λ ≤ δ } where z u,cs ( t ) ∈ Y u,cs ( t ) and | z | λ , sup t ≤ e − λt | z ( t ) | W k,q . This Γ is the set of functions with the desired backward in time decay expected to be satisfiedby the solutions on the unstable manifolds. For any z ∈ Γ and(2.34) z u ∈ Y u (0) , | z u | W k,q ≤ δ C where C is given in Lemma 2.6, define L ( · , z u ), where ˜ z = (˜ z u , ˜ z cs ) = L ( z, z u ) is definedas, for any t ≤ ( ˜ z u ( t ) = T u ( t, z u + R t T u ( t, τ ) F u (cid:0) τ, z ( τ ) (cid:1) dτ ˜ z cs ( t ) = R t −∞ T cs ( t, τ ) F cs (cid:0) τ, z ( τ ) (cid:1) dτ. Proposition 2.7.
There exists
K > depending only on r and k such that if λ and δ satisfy λ ∈ (cid:0) λ cs + Kµ, λ u − Kµ (cid:1) (2.36) C C δ (cid:0) λ − λ cs + 1 λ u − Kµ − λ (cid:1) < where C and C are from Lemma 2.6 and (2.33) , respectively, then L ( · , z u ) is a contractionon Γ with the Lipschitz constant for any z u satisfying (2.34) . Moreover |L| C r − , ≤ C forsome C > depending only on r, k, q, n, and v . Remark.
Assumption (A3) with a reasonably large K , depending only on r and k , guaranteesthe existence of λ and δ satisfying the above inequalities. Remark.
It is standard in the invariant manifold theory (see, for example, [CL88] ) to provethat z ∈ Γ solves (2.29) with z u (0) = z u if and only if, z is the fixed point of L ( · , z u ) .Therefore, Proposition 2.7 shows that, for any given z u satisfying (2.34) , there exists aunique solution of (2.29) satisfying the exponential decay as t → −∞ with the decay rate atleast λ .Proof. In the proof the generic constant K may change from line to line, but always dependsonly on k and r . From the definition of L , Lemma 2.6, assumptions (A3), (2.36), and (2.34),and the second order Taylor expansion of F based on (2.33), (instead of the usual smallLipschitz estimates of F ), we obtain for t ≤ e − λt | ˜ z u ( t ) | W k,q ≤ C | z u | W k,q + Z t C C e − λt +( λ u − Kµ )( t − τ ) − Kµτ +2 λτ dτ | z | λ ≤ C | z u | W k,q + 12 C C | z | λ Z t e ( λ u − Kµ − λ )( t − τ ) dτ ≤ C | z u | W k,q + C C δ λ u − Kµ − λ ) | z | λ and e − λt | ˜ z cs ( t ) | W k,q ≤ Z t −∞ C C e − λt + λ cs ( t − τ ) − Kµτ +2 λτ dτ | z | λ ≤ C C δ | z | λ λ − λ cs ) . Therefore (2.37) implies that(2.38) |L ( z, z u ) | λ ≤ C | z u | W k,q + 14 | z | λ ≤ δ and thus L ( · , z u ) maps Γ into itself.To prove L ( · , z u ) is a contraction on Γ , we note that for any z , ∈ Γ , (2.33) implies | F u,cs ( t, z ( t )) − F u,cs ( t, z ( t )) | ≤ C δ e (2 λ − Kµ ) t | z − z | λ , ∀ t ≤ e − λt | ˜ z u ( t ) − ˜ z u ( t ) | ≤ C C δ Z t e ( λ u − Kµ − λ )( t − τ ) dτ | z − z | λ ≤ C C δ | z − z | λ λ u − Kµ − λe − λt | ˜ z cs ( t ) − ˜ z cs ( t ) | ≤ C C δ Z t −∞ e ( λ cs − λ )( t − τ ) dτ | z − z | λ ≤ C C δ | z − z | λ λ − λ cs and thus (2.37) implies that L ( · , z u ) is a contraction.Since L is linear in z u , we only need to prove its smoothness in z ∈ Γ . For any z ∈ Γ ,formally the linearization of L is given by D z L ( z, z u ) z = ¯ z = (¯ z u , ¯ z cs )where for t ≤ ( ¯ z u ( t ) = R t T u ( t, τ ) D z F u (cid:0) τ, z ( τ ) (cid:1) z ( τ ) dτ ¯ z cs ( t ) = R t −∞ T cs ( t, τ ) D z F cs (cid:0) τ, z ( τ ) (cid:1) z ( τ ) dτ. The same procedure as in the above shows that |D z L| λ ≤ . To show it is indeed thederivative of L , take z , ∈ Γ , let˜ z , = (˜ z , u , ˜ z , cs ) = L ( z , , z u ) , ¯ z = (¯ z u , ¯ z cs ) = D z L ( z , z u )( z − z ) . NSTABLE MANIFOLDS 19
It is straight forward to compute from (2.33) and Lemma 2.6, that for t ≤ e − λt | ˜ z u ( t ) − ˜ z u ( t ) − ¯ z u ( t ) | = e − λt | Z t T u ( t, τ ) (cid:16) F u (cid:0) τ, z ( τ ) (cid:1) − F u (cid:0) τ, z ( τ ) (cid:1) − D F u (cid:0) τ, z ( τ ) (cid:1)(cid:0) z ( τ ) − z ( τ ) (cid:1)(cid:17) dτ |≤ C C | z − z | λ Z t e − λt +( λ u − Kµ )( t − τ ) − Kµτ +2 λτ dτ ≤ C C | z − z | λ λ u − Kµ − λ . The estimate for the center-stable component is very similar and this proves that D z L isindeed the derivative of L .Finally, we will show that D z L is Lipschitz. In fact, let¯ z , = (¯ z u , ¯ z cs ) = D z L ( z , , z u ) z. Then for any t ≤ e − λt | ¯ z cs ( t ) − ¯ z cs ( t ) | = e − λt | Z t −∞ T cs ( t, τ ) (cid:0) D z F cs ( τ, z ( τ )) − D z F ( τ, z ( τ )) (cid:1) z ( τ ) dτ |≤ C C | z − z | λ | z | λ Z t −∞ e ( λ cs − λ )( t − τ )+( λ − Kµ ) τ dτ ≤ C C λ cs − λ | z − z | λ | z | λ . The estimates for the unstable component is the same and the proof of the higher ordersmoothness is similar. (cid:3)
From the Contraction Mapping Theorem and Proposition 2.7, the mapping L has a uniquefixed point z ∗ ( t, z u ) = (cid:0) z ∗ u ( t, z u ) , z ∗ cs ( t, z u ) (cid:1) which is C r − , in z u and satisfies z ∗ u (0 , z u ) = z u . Moreover, (2.38) implies(2.40) | z ∗ ( u ) | λ ≤ C | z u | W k,q . Like in the standard invariant manifold theory, these are solutions on the invariant integralunstable manifold of the non-autonomous system (2.25). Define h L ( z u ) = z ∗ cs (0 , z u ) ∈ X cs for all z u satisfying (2.34), then the C r − , graph W uL , graph ( h L )defines the slice of the unstable integral manifold of (2.25) for t = 0. Obviously the unique-ness of the fixed point implies that z ∗ ( · ,
0) = 0 and thus h L (0) = 0 and 0 ∈ W uL . Differenti-ating the fixed point equation we obtain D z u z ∗ ( z u ) = D z u L (cid:0) z ∗ ( z u ) , z u (cid:1) + D z L (cid:0) z ∗ ( z u ) , z u (cid:1) D z u z ∗ ( z u )Clearly, (2.39) implies that D z L (0 , z u ) = 0 and thus D z u z ∗ ( · ,
0) = D z u L (0 ,
0) = (cid:0) T u ( · , , (cid:1) which does not have the center-stable component. Therefore, we obtain that D z u h L (0) = 0which means that, at 0, the tangent space of the unstable integral manifold W uL T W uL = Y u (0) = { ( U ( v ) , v ) T | v ∈ X u } , where(2.41) U ( v ) , Z −∞ (cid:0) Du ( τ ) (cid:1) − (cid:0) ( e τL v ) ◦ u ( τ ) (cid:1) dτ, | U | L ( X u ,W k,qEuler ) ≤ ∞ . Here the boundedness of U follows from (2.3), (2.36) and assumptions (A2) and (A3).2.6. Unstable manifold in the Eulerian coordinates.
From the unstable integral man-ifold (at t = 0) W uL constructed in the Lagrangian coordinates and the corresponding rela-tionship given in (2.22) and (2.24), we obtain the C r − , invariant unstable manifold in theEulerian coordinates W u , { v = v + (cid:0) D Ψ( w ) w (cid:1) ◦ Ψ( w ) − | ( w, w ) ∈ W uL } . The above expression was derived by substituting t = 0 into (2.24). Lemma 2.8.
There exist
K > depending only on r and k ) , δ , C > depending only on r, k, q, n , and v such that (1) There exists H ∈ C r − , (cid:0) B δ ( X u ) , X cs (cid:1) satisfying | H | C r − , ≤ C , H (0) = 0 , DH (0) =0 , and { v + v + H ( v ) | v ∈ B δ ( X u ) } ⊂ W u ∩ (cid:0) v + B δ ( W k,qEuler ) (cid:1) ⊂ { v + v + H ( v ) | v ∈ B δ ( X u ) } . (2) For any v ∈ (cid:0) v + B δ ( W k,qEuler ) (cid:1) ∩ W u the solution v ( t ) of the Euler equation (E)with the initial value v (0) = v satisfies v ( t ) ∈ W u , | v ( t ) − v | W k,q ≤ C | v − v | W k,q e ( λ − Kµ ) t , ∀ t ≤ . Proof. (1) For any v ∈ X u with | v | W k,q < δ C (cid:0) | U | L ( X u ,W k,qEuler ) (cid:1) , let z u = ( U v , v ) ∈ Y u (0), where U is defined in (2.41) and it implies z u satisfies (2.34),and G ( v ) = (cid:0) D Ψ( w ) w (cid:1) ◦ Ψ( w ) − where ( w, w ) = z ∗ (0 , z u ) = z u + h L ( z u ) ∈ W uL . Clearly the definition of W uL and W u and the properties of h L imply W u = { v + G ( v ) } and G (0) = 0 , DG (0) v = w = v , and G ∈ C r − , with bounds depending only on r, k, q, n , and v . Therefore, the existenceand properties of H follows from the Implicit Function Theorem immediately.(2) For any v = v + (cid:0) D Ψ( w ) w (cid:1) ◦ Ψ( w ) ∈ (cid:0) v + B δ ( W k,qEuler ) (cid:1) ∩ W u , with ( w , w ) ∈ W uL , let v ( t ) be the solution of (E) with v (0) = v and z ( t ) = (cid:0) w ( t ) , w t ( t ) (cid:1) the solution of (2.25)with the initial value z (0) = ( w , w ). Since z (0) = ( w , w ) ∈ W uL and thus z ∈ Γ and(2.40) implies | w ( t ) | W k,q + | w t ( t ) | W k,q ≤ e λt | z | λ ≤ C e λt | z u | W k,q , t ≤ . Therefore (2.24) implies the desired decay estimate. | v ( t ) − v | W k,q ≤ Ce ( λ − Kµ ) t → t → −∞ . NSTABLE MANIFOLDS 21
To see the local invariance of W u , fixed T ≥ , for t ∈ ( −∞ , T ], let˜ w ( t ) = Ψ − (cid:16) u ( t ) ◦ Ψ (cid:0) w ( t + t ) (cid:1) ◦ u ( t ) − (cid:17) , t ≤ . Since the right translation invariance and (2.22) imply˜ u ( t ) = u ( t ) ◦ Ψ (cid:0) ˜ w ( t ) (cid:1) = u ( t + t ) ◦ Ψ (cid:0) w ( t + t ) (cid:1) ◦ u ( t ) − = u ( t + t ) ◦ u ( t ) − is a solution of (2.14), we have ( ˜ w, ˜ w t ) is a solution of (2.25) which clearly corresponds tothe solution ˜ v ( t ) = v ( t + t ) of the Euler equation (E). Moreover, Proposition 2.1 implies | ˜ w ( t ) | W k,q + | ˜ w t ( t ) | W k,q ≤ C (cid:0) | w ( t + t ) | W k,q + | w t ( t + t ) | W k,q (cid:1) ≤ Ce λ ( t + t ) | z u | W k,q ≤ Ce λT (cid:0) | U | L ( X u ,W k,qEuler ) (cid:1) | v − v | W k,q e λt ≤ δ Ce λT (cid:0) | U | L ( X u ,W k,qEuler ) (cid:1) e λt . By choosing δ ≤ δ Ce λT (cid:0) | U | L ( X u ,W k,qEuler ) (cid:1) , we obtain that the solution ˜ z ( t ) = (cid:0) ˜ w ( t ) , ˜ w t ( t ) (cid:1) ∈ Γ . Therefore v ( t ) = ˜ v (0) ∈ W u whichimplies the invariance. (cid:3) The property DH (0) = 0 immediately implies, as expected, the tangent space at thesteady state v is given by T v W u = X u and the proof of Theorem 1.1 is complete.3. Two-dimensional Euler equations
In this and the next sections, we will illustrate how assumptions (A1) – (A3) can besatisfied for certain steady states of the Euler equation (E). In this section, we consider thecase of Ω = S × ( − y , y ), that is, 2 π − periodic in x and with rigid walls on { y = ± y } .Let v = ( v , v ) T : Ω → R satisfy ∇ · v = 0 in Ω and v · N = 0 on ∂ Ω. On the one hand,let ω = ∂ x v − ∂ y v , s = 1 | Ω | Z Ω v dxdy be the curl and the average horizontal momentum, respectively. We note that s is an invariantof (E) due to the translation symmetry in x . On the other hand, v is uniquely determinedby ω and s through(3.1) v = J ∇ ∆ − ω + se , J = (cid:18) −
11 0 (cid:19) , e = (1 , T where ∆ − is the inverse of the Laplacian with zero Dirichlet boundary condition. It is clearthat 1 C | v | W k,q ≤ | ω | W k − ,q + | s | ≤ C | v | W k,q , k ≥ , q > C >
0. In the ( ω, s ) representation, (E) takes the form(3.2) ω t + v · ∇ ω = 0 , s t = 0where v is considered as determined by ( ω, s ) by (3.1). Remark.
Due to the nontrivial first cohomology group of Ω , the vorticity alone does notdetermine a vector field in W k,qEuler and thus the average horizontal momentum has to beincluded in the reformulation of the problem. If one considers Ω = T , the D torus, thenboth momentum invariants each of which corresponds to a nontrivial element in the firstcohomology groups should be included.
Suppose v ∈ W k +4 ,q , k > q , is a steady state of (E), which corresponds to ( ω , s )has Lyapunov exponent µ ≥ ω , s ) to obtain(3.3) (cid:18) ωs (cid:19) t = − (cid:18) v · ∇ ω (cid:19) − (cid:18) ( J ∇ ∆ − ω + se ) · ∇ ω (cid:19) , L (cid:18) ωs (cid:19) + L (cid:18) ωs (cid:19) . Assume that there exists an unstable eigenvalue λ with Re λ > ( k − µ of the lin-earized Euler operator L (defined in (1.2)) on L q , and let v ∈ L q be the eigenfunction withcorresponding ω and s . Then obviously s = 0 and λ ω + v · ▽ ω = − v · ∇ ω . An integration of above along the steady trajectory X ( s ) yields ω = Z ∞ e − λ s v · ∇ ω ( X ( s )) ds. By the standard bootstrap argument and the assumption Re λ > ( k − µ , we get ω ∈ W k − ,q and v ∈ W k,q .For any λ − ∈ (( k − µ , Re λ ) which does not equal the real part of any unstable eigen-value , let ˜ X cs = { ( ω, s ) T | ω ∈ W k − ,q | lim sup 1 t log | e t ( L + L ) ω | W k − ,q ≤ λ − } which is clearly a invariant subspace of e L + L . Let σ cs = σ ( e L + L | ˜ X cs ) , σ u = σ ( e L + L | W k − ,q × R ) \ σ cs , λ + = log(inf {| λ | | λ ∈ σ u } ) ≥ λ − . As the groups of bounded operators e t ( L + L ) and e tL are conjugate through (3.1), we alsohave the invariance of X cs,u under e tL . Since e tL ω = ω ◦ u ( t ) − , inequality (2.3) implies | e tL | L ( W k − ,q ) ≤ Ce ( k − µ | t | for any µ > µ and some C > µ . In particular, v is divergence free, yields that e tL is a group of isometries on any L q space. Since L is acompact operator acting on ( ω, s ) T , e t ( L + L ) is a compact perturbation to e tL in the space W k − ,q and thus • λ + > λ − and σ u is an isolated compact subset of σ ( e L + L ). Let ˜ X u be the eigenspaceof e L + L corresponding to σ u , and X cs,u = { v ∈ W k,qEuler | ( ω, s ) ∈ ˜ X cs,u } . • ˜ X u and X u are finite dimensional and • (A2) is satisfied for any λ cs and λ u with λ − < λ cs < λ u < λ + .Assumption (A3) depends on the Lyapunov exponents of v . In particular, if v is alinearly unstable shear flow, µ = 0 and (A3) is also satisfied. An example is v = (sin βy, v is linearly unstable when β > (cid:16) π y (cid:17) < β − NSTABLE MANIFOLDS 23
Remark.
Consider rotating flows v = U ( r ) ~e θ in an annulus Ω = { a < r < b } . Then bysimilar arguments as above, assumptions (A2)-(A3) are satisfied as long as v is linearlyunstable. Three-dimensional Euler equations
In this section, we construct examples of 3D unstable steady flows for which Theorem 1.1can be applied to get unstable (stable) manifolds. Consider Ω = T to be a 3D torus withperiods L x , L y and L z in x, y and z variables. For any profile U ( y, z ) , the 3D shear flow ~u = ( U ( y, z ) , ,
0) is a steady solution of 3D Euler equation. We construct unstable 3Dshears satisfying assumptions (A1)-(A3) in several steps.The linearized 3D Euler equation around a 3D shear ( U ( y, z ) , ,
0) is(4.1) ∂ t u + U u x + vU y + wU z = − P x , (4.2) ∂ t v + U v x = − P y , ∂ t w + U w x = − P z , (4.3) u x + v y + w z = 0 , with periodic boundary conditions. There are almost no results about the the linear insta-bility of general 3D shears. So we construct unstable 3D shears near unstable 2D shear flows( U ( y ) , ,
0) where U ( y ) is periodic with period L y . First, we give a sufficient condition forlinear instability of 2D periodic shears, which generalizes the result in [L03] for shear flowsin a channel with rigid walls. Lemma 4.1.
Consider a periodic shear profile U ( y ) ∈ C (0 , L y ) with only one inflectionvalue U s and (4.4) K ( y ) = − U ′′ ( y ) U ( y ) − U s > . Let − α be the lowest eigenvalue of the Sturm-Liouville operator (4.5) Lϕ = − ϕ ′′ − K ( y ) ϕ with the periodic boundary conditions on y ∈ [0 , L y ] . Then the Rayleigh equation (4.6) U ′′ φ − ( U − c ) (cid:0) φ ′′ − α φ (cid:1) = 0 , with periodic boundary conditions on y ∈ [0 , L y ] has unstable eigenmodes ( Im c > ) for any α ∈ (0 , α max ) . Remark.
Under the assumptions in the above Lemma, the lowest eigenvalue of L is negative,since ( L (1) ,
1) = − R K ( y ) dy < . A typical example satisfying (4.4) is U ( y ) = sin (cid:16) πL y y (cid:17) for which K ( y ) = (cid:16) πL y (cid:17) .Proof. The proof is similar to the case of rigid walls ([L03]), so we only point out somesmall modifications. Let φ s be the eigenfunction of L corresponding to the lowest eigenvalue − α . Then ( φ, c, α ) = ( φ s , U s , α max ) is a neutral solution to the Rayleigh equation (4.6).By Sturm-Liouville theory, − α is a simple eigenvalue and we can take φ s >
0. First, westudy bifurcation of unstable modes near the neutral mode. Denote y to be a minimum point of φ s and let y = y + L y . We normalize φ s such that φ s ( y ) = 1 , φ ′ s ( y ) = 0. Define φ ( y ; c, ε ) and φ ( y ; c, ε ) to be the solutions of(4.7) − φ ′′ + U ′′ U − U s − c φ + (cid:0) α + ε (cid:1) φ = 0 , with φ ( y ) = 1 , φ ′ ( y ) = 0 and φ ( y ) = 0 , φ ′ ( y ) = 1 . Here ε < c > . Define I ( c, ε ) = φ ( y ; c, ε ) + φ ′ ( y ; c, ε ) − , then the existence of a solution to the Rayleigh equation (4.7) with periodic boundaryconditions on y ∈ [ y , y ] is equivalent to the existence of a root of I with Im c > . When c → , ε → − and | Re c | / Im c remains bounded, as in [L03] we can show that φ ( y ; c, ε )( φ ( y ; c, ε )) converges to φ s ( y ) ( φ z ( y )) uniformly in C [ y , y ]. Here, φ z ( y ) ∈ C [ y , y ]satisfies that φ ′ z ( y ) = 1 and φ z ( y ) = 0 since φ z ( y ) can not be another eigenfunctionassociated with the simple eigenvalue − α . By similar calculations as in [L03], it can beshown that when c → , ε → − and | Re c | / Im c remains bounded, ∂I∂ε → φ z ( y ) Z y y φ s ( y ) dy, and ∂I∂c → − φ z ( y ) iπ l X k =1 (cid:16) | U ′ | − Kφ s (cid:17) | y = a k + P Z y y (cid:0) K ( y ) φ s ( y ) (cid:1) / ( U ( y ) − U s ) dy ! .Here, a , · · · , a l are the inflection points such that U ( a k ) = U s , k = 1 , · · · , l and P R y y denotes the Cauchy principal part. Then by a variant of implicit function theorem as in[L03], there exists ε < ε < ε < φ ε with c = c ( ε ) to Rayleigh’s equation (4.7). By the same arguments in [L03], such unstable modescan be continuated to all wave numbers α ∈ (0 , α max ). (cid:3) Our second step is to show that 3D shears near an unstable 2D shear are also linearlyunstable. More precisely, we have
Lemma 4.2.
Let U ( y ) ∈ C (0 , L y ) be such that the Rayleigh equation (4.6) has an unstablesolution with ( α , c ) ( α , Im c > . Fixed L z > , consider U ( y, z ) ∈ C ((0 , L y ) × (0 , L z )) which is L y , L z -periodic in y and z respectively. If k U ( y, z ) − U ( y ) k W ,p ((0 ,L y ) × (0 ,L z )) ( p > issmall enough, then there exists an unstable solution e iα ( x − ct ) ( u, v, w, P ) ( y, z ) to the lin-earized equation (4.1)-(4.3) with | c − c | small. Moreover, if U ( y, z ) ∈ C ∞ , then ( u, v, w, P ) ∈ C ∞ . The proof of above lemma is almost the same as in the case of rigid walls ([LL11]), sowe skip it here. By Lemmas 4.1 and 4.2, there exist linearly unstable 3D shears ~u =( U ( y, z ) , , W m, Euler = H m ( m ≥ µ = 0. Let G t = e Lt be the linearized Euler semigroupnear a steady flow ~u ( ~x ) and denote r ess ( G t ; H m ) to be the essential spectrum radius of G t in space H m . By rather standard semigroup theory (see e.g. [Shi83, Section 1 ]), to get thelinear exponential dichotomy (A2), it suffices to show that r ess ( G t ; H m ) = 1. The essentialspectrum of linearized Euler operator had been studied a lot ([FV91] [LM91] [SL09] [V96])by using the geometric optics method. We use the following characterization of r ess ( G t ; H m )in [SL09]. NSTABLE MANIFOLDS 25
Lemma 4.3. [SL09]
Consider the following ODE system (4.8) ~x t = ~u ( ~x ) ~ξ t = − ∂~u ( ~x ) T ~ξ~b t = − ∂~u ( ~x ) ~b + 2 (cid:16) ∂~u ( ~x ) ~b, ~ξ (cid:17) ~ξ (cid:12)(cid:12)(cid:12) ~ξ (cid:12)(cid:12)(cid:12) − , where ~u ( ~x ) is a steady flow of 3D Euler equation in T and ~x ∈ T , ~ξ,~b ∈ R . Denote (4.9) Λ m = lim t →∞ t ln sup ~x ∈ T , | ~ξ | =1 ~b ⊥ ~ξ , | ~b | =1 (cid:12)(cid:12)(cid:12) ~b ( t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) m where (cid:16) ~x ( t ) ,~b ( t ) , ~ξ ( t ) (cid:17) is the solution of (4.8) with initial data (cid:16) ~x , ~ξ ,~b (cid:17) . Then we have r ess ( G t ; H m ) = e t Λ m . Lemma 4.4.
For ~u = ( U ( y, z ) , , in T , we have Λ m = 0 .Proof. Denote ~x ( t ) = ( x ( t ) , y ( t ) , z ( t )) , ~ξ ( t ) = ( ξ ( t ) , ξ ( t ) , ξ ( t )) , ~b ( t ) = ( b ( t ) , b ( t ) , b ( t )) , and ~x = ( x , y , z ) , ~ξ = (cid:0) ξ , ξ , ξ (cid:1) , ~b = (cid:0) b , b , b (cid:1) . The solution of first two equations of (4.8) yield x ( t ) = x + U ( y , z ) t, y ( t ) = y , z ( t ) = z and ξ ( t ) = ξ , ξ ( t ) = − U y ξ t + ξ , ξ ( t ) = − U z ξ t + ξ . Plugging above forms into the equation of ~b ( t ), we have(4.10) ˙ b = − ( U y b + U z b ) + 2 ( ξ ) ( U y b + U z b ) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) , (4.11) ˙ b = 2 ξ ( U y b + U z b ) ( ξ − U y ξ t ) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) , (4.12) ˙ b = 2 ξ ( U y b + U z b ) ( ξ − U z ξ t ) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) . To show that Λ m = 0, it suffices to prove that (cid:12)(cid:12)(cid:12) ~b ( t ) (cid:12)(cid:12)(cid:12) only has polynomial growth, uniformlyin (cid:16) ~x , ~ξ ,~b (cid:17) . From equations (4.11) and (4.12), we have ddt ( U y b + U z b )= 2 ( U y b + U z b ) h ξ ξ U y + ξ ξ U z − ( ξ ) (cid:0) U y + U z (cid:1) t i(cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) = − ( U y b + U z b ) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) , by noting that (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) = (cid:0) ξ (cid:1) + (cid:0) − U y ξ t + ξ (cid:1) + (cid:0) − U z ξ t + ξ (cid:1) = 1 − (cid:0) ξ ξ U y + ξ ξ U z (cid:1) t + (cid:0) ξ (cid:1) (cid:0) U y + U z (cid:1) t . Thus ddt (cid:20) ( U y b + U z b ) (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) (cid:21) = 0and(4.13) ( U y b + U z b ) ( t ) = U y b + U z b (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) . For any fixed t >
0, we find a lower bound for (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) by minimizing the function f (cid:0) ξ , ξ , ξ (cid:1) = (cid:0) ξ (cid:1) + (cid:0) − U y ξ t + ξ (cid:1) + (cid:0) − U z ξ t + ξ (cid:1) subject to the constraint ( ξ ) + ( ξ ) + ( ξ ) = 1. By calculations of Lagrange multiplier,we get min | ~ξ | =1 (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) = 2 + (cid:0) U y + U z (cid:1) t − q(cid:0) (cid:0) U y + U z (cid:1) t (cid:1) −
42= 22 + (cid:0) U y + U z (cid:1) t + q(cid:0) (cid:0) U y + U z (cid:1) t (cid:1) − t >
0, we get the estimate (cid:12)(cid:12)(cid:12) ~ξ ( t ) (cid:12)(cid:12)(cid:12) ≥
12 + (cid:0) U y + U z (cid:1) t . Thus by (4.13), | ( U y b + U z b ) ( t ) | ≤ c t + c , for c , c > (cid:16) ~x , ~ξ ,~b (cid:17) . By (4.10)-(4.12), we have (cid:12)(cid:12)(cid:12) ˙ b (cid:12)(cid:12)(cid:12) ≤ | ( U y b + U z b ) ( t ) | , (cid:12)(cid:12)(cid:12) ˙ b (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ˙ b (cid:12)(cid:12)(cid:12) ≤ | ( U y b + U z b ) ( t ) | , NSTABLE MANIFOLDS 27 and thus (cid:12)(cid:12)(cid:12) ~b ( t ) (cid:12)(cid:12)(cid:12) ≤ c t + c for some constants c , c >
0. This finishes the proof of the lemma. (cid:3)
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Translations of Math-ematical Monographs vol 74 (1989), American Mathematical Society. † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332
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